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Circles
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Area to radius to circumference
- Start:
- Write the given area or circumference as πr² or 2πr.
- Stop:
- Stop when you have r as a number.
- Skip:
- Skip decimals for π.
- Why:
- Answers are in terms of π, so keep π and compare.
Used in question 1
2. Flip the signs for the centre
- Start:
- Look inside each bracket (x − 3) and (y + 2).
- Stop:
- Stop after writing h and k with the opposite signs.
- Skip:
- Skip the right side until you need the radius.
- Why:
- The formula has a minus in front of h and k, so a plus in the bracket means a negative coordinate.
Used in questions 2, 6
3. Fraction of 360 first
- Start:
- Write angle ÷ 360 as a simple fraction.
- Stop:
- Stop when you have it reduced (1/4, 1/6).
- Skip:
- Skip the radius if the question gives arc and circumference.
- Why:
- The slice is always the same fraction of the whole, whether length or area, as in the 6π ÷ 36π question.
Used in questions 3, 4, 7
4. Right side is r squared
- Start:
- Find the number after the equals sign.
- Stop:
- Stop and take its square root only if the question asks for r.
- Skip:
- Do not square or root again when matching an equation.
- Why:
- Many wrong options use 25 as r or 5 as r².
Used in questions 5, 6, 8
5. Complete the square quickly
- Start:
- Halve the x-coefficient and the y-coefficient and square them.
- Stop:
- Stop when both brackets are perfect squares.
- Skip:
- Do not expand the answer options.
- Why:
- It converts any messy circle equation to the readable form in two quick steps.
Used in questions 5, 8
6. Use midpoints for a diameter
- Start:
- Average the x values and the y values.
- Stop:
- Stop when you have the centre and the diameter squared.
- Skip:
- Skip the full distance square root: divide the squared distance by 4 for r².
- Why:
- Radius squared is diameter squared over 4, which avoids a square root.
Used in question 6
7. Radius meets tangent at 90°
- Start:
- Mark the radius that goes to the point where the line touches the circle.
- Stop:
- Stop when you write a right angle there and name the right triangle.
- Skip:
- Skip any angle you cannot connect to that triangle.
- Why:
- A tangent is always perpendicular to the radius, so Pythagoras finishes question 11. A diameter works the same way, because any angle on a semicircle is 90° (question 12).
Used in questions 11, 12
8. Distance from centre gives radius
- Start:
- For a circle tangent to an axis or a line, find the distance from the centre to that line.
- Stop:
- Stop when you have r and then r².
- Skip:
- Skip expanding the brackets of the answer options.
- Why:
- Tangent means the distance equals the radius. For centre (4, −3) the y-axis is 4 away, so r² = 16; the x-axis is 3 away, which gives the trap 9. The highest point of a circle is centre y + r.
Used in questions 14, 15
9. Chord splits into a right triangle
- Start:
- Draw the line from the centre to the middle of the chord.
- Stop:
- Stop when you know half the chord, the radius and the new distance, a right triangle.
- Skip:
- Skip trig and angles.
- Why:
- That line is perpendicular to the chord and cuts it in half, so a 6-8-10 triple gives the height 6 and the area ½ × 16 × 6 = 48.
Used in questions 16, 11